Lagrangean/Surrogate Heuristics for p-Median Problems

Abstract

Abstract:The p-median problem is the problem of locating p facilities (medians) on a network so as to minimize the sum of all the distances from each demand point to its nearest facility. A successful approach to approximately solve this problem is the use of Lagrangean heuristics, based upon Lagrangean relaxation and subgradient optimization. The Lagrangean/surrogate is an alternative relaxation proposed recently to correct the erratic behavior of subgradient like methods employed to solve the Lagrangean dual. We propose in this paper Lagrangean/surrogate heuristics to p-median problems. Lagrangean and surrogate relaxations are combined relaxing in the surrogate way the assignment constraints in the p-median formulation. Then, the Lagrangean relaxation of the surrogate constraint is obtained and approximately optimized (one-dimensional dual). Lagrangean/surrogate relaxations are very stable (low oscillating) and reach the same good results of Lagrangean (alone) heuristics in less computational times. Two primal heuristics was tested, an interchange heuristic and a location-allocation based heuristic. The paper presents several computational tests which have been conducted with problems from the literature, a set of instances presenting large duality gaps, a set of time consuming instances and a large scale instance.

Constructive Genetic Algorithm for Clustering Problems

“…The Lagrangian heuristic results appeared in Senne and Lorena (2000); they used local search and an iterative locationallocation procedure to obtain feasible solutions. It was programmed in C, running on an IBM Risc/6000 model 3AT workstation (compiled using xlc compiler with -O2 optimization option).…”

“…A classical mathematical programming formulation for p-median problems has been considered (Beasley, 1985(Beasley, , 1993Christofides and Beasley, 1982;Senne and Lorena, 2000). Considering a primal-dual approach, the set of OR-Library p-median instances can be considered that is composed of easy problems in the sense of duality gaps.…”

“…Considering a primal-dual approach, the set of OR-Library p-median instances can be considered that is composed of easy problems in the sense of duality gaps. The gaps can all be closed using a Lagrangian (or Lagrangian/surrogate) approach (Beasley, 1993; Senne and Lorena, 2000). Based on that, another test suit was performed with another set of instances (Galvão and ReVelle, 1996;Senne and Lorena, 2000) that presents large duality gaps for some values of p. A comparison was also made with the direct application of the interchange heuristic IH on a set of structures.…”

Parallelization of the scatter search for the p-median problem

“…Heuristics based on linear programming have been proposed by du Merle at al. [4] and by Senne and Lorena [37,38]. Another type of heuristics suggested in the literature is based on the relaxed dual of the integer programming formulation of the p-median problem found in the papers by Erlenkotter [5], Galvã ao [7] and Nauss and Markland [28].…”

New Genetic Algorithms Based Approaches to Continuous p-Median Problem

“…Such approaches cannot always produce an optimal solution in reasonable time. Some researches on facility location have been conducted using traditional heuristics, for example, Goldengorin et al (2004) developed a branch and peg algorithm that enhance the performance of branch-and-bound algorithm for the simple plant location problem; Senne and Lorena (1999) proposed a Lagrangean and surrogate heuristic approach to p-median problems; Revelle and Swain (1970) provided the first linear programming formulation of the p-median problem; Lorena and Senne (2003) explored the application of column generation to a capacitated p-median location problem; Xu (1999) solved a joint inventory-location problem by a substitution algorithm; Okabe et al (1992) illustrated the usefulness of Voronoi diagrams to solve locational optimization problems.…”